Quadrupole shape phase transitions in the γ–rigid regime

Petrică Buganu, and Radu Budaca

IFIN-HH, Bucharest – Magurele, Romania

International Workshop “Shapes and Dynamics of Atomic Nuclei: Contemporary Aspects” (SDANCA – 15), 8 – 10 October 2015, Sofia, Bulgaria

The Bohr-Mottelson Hamiltonian:

The γ-rigid Hamiltonian for γ=30o:

The γ-rigid Hamiltonian for γ=0o:

2 2 2 234

4 22 21

1 2 3 1 2 3 1 2 3

1 1sin 3 ,

22 2 sin 3 8 sin ( )3

, , , , , , , , , , , Euler angles

k

k

QH V

B B B k

H E

2 23 2 2

13 2

1 2 3 1 2 3

1 3ˆ ˆ2 2 4

, , , , , ,

H Q Q VB B

H E

2 2 22

2 2 2 2

1 1 1sin

2 6 sin sin

, , , ,

H VB B

H E

E(5): F. Iachello, Phys. Rev. Lett. 85 (2000) 3580. spherical vibrator to γ-unstable rotorX(5): F. Iachello, Phys. Rev. Lett. 87 (2001) 052502. spherical vibrator to axial rotorY(5): F. Iachello, Phys. Rev. Lett. 91 (2003) 132502. axial rotor to triaxial rotorZ(5): D. Bonatsos, D. Lenis, D. Petrellis, and P. A. Terziev, Phys. Lett. B 588 (2004) 172. prolate rotor to oblate rotor?!

Z(4): D. Bonatsos, D. Lenis, D. Petrellis, P. A. Terziev, and I. Yigitoglu, Phys. Lett. B 621 (2005) 102. A. S. Davydov, and A. A. Chaban, Nucl. Phys. 20 (1960) 499.

X(3): D. Bonatsos, D. Lenis, D. Petrellis, P. A. Terziev, and I. Yigitoglu, Phys. Lett. B 632 (2006) 238.

A. Bohr, Mat. Fyz. Medd. K. Dan. Vidensk. Selsk. 26 (1952) No. 14.A. Bohr, and B. R. Mottelson, Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 27 (1953) No. 16.

The potentials in the β variable and the γ rigidityvalues for the most recent γ-rigid solutions.

D. Bonatsos, D. Lenis, D. Petrellis, P. A. Terziev, and I. Yigitoglu, Phys. Lett. B 621 (2005) 102.

D. Bonatsos, D. Lenis, D. Petrellis, P. A. Terziev, and I. Yigitoglu, Phys. Lett. B 632 (2006) 238.

R. Budaca, Eur. Phys. J. A 50 (2014) 87.

R. Budaca, Phys. Lett. B 739 (2014) 56.

P. Buganu, and R. Budaca, J. Phys. G: Nucl. Part. Phys. 42 (2015) 105106.

P. Buganu, and R. Budaca, Phys. Rev. C 91 (2015) 014306.

Sextic oscillator potential

2 2 4 2 614 2 , 0,1,2,...

2

1 3 2 2

2 2

1 . , , ( 1, 4),( 2, 8),...

2

v b a s M ab a M

s s

s M const c M L M L M L

2

2 2 2 2

2

2 2, ,

( 1) 3 for X 3 -Sextic and ( 1) ( 1) for Z(4)-Sextic

3 4

d B Bv v V E

d

L LL L R

Exact separation of the variables:

X(3)-Sextic and Z(4)-Sextic

4 2

22 2 4 2 6

2 2

122 2 4 2

22 2 2

2

1 32 2

12 2 4 2 ,2

, Ansatz: ,

4 1, 2 2 2 2 .

a bs

M

M M

s sH b a s M ab a

H NP e

sQP P Q b s a M

The quasi-exactly solution for the sextic potential

A. G. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics, (Institute of Physics Publishing, Bristol, 1994)

Numerical results

Z(4)-Sextic X(3)-Sextic

1

4

2 2 4 6

0,0,0

022 0,0,0

0,0

02 0,0

1, ,

2

4 2

2c

nLRnLR

nLnL

by a c s M

a

v y c y y y

c

E ER

E E

E ER

E E

Z(4)-Sextic: P. Buganu, and R. Budaca, Phys. Rev. C 91 (2015) 014306.X(3)-Sextic: P. Buganu, and R. Budaca, J. Phys. G: Nucl. Part. Phys. 42 (2015) 105106.

Z(4)-Sextic: P. Buganu, and R. Budaca, Phys. Rev. C 91 (2015) 014306.X(3)-Sextic: P. Buganu, and R. Budaca, J. Phys. G: Nucl. Part. Phys. 42 (2015) 105106.

Degenerate states!A possible dynamical symmetry?!

Z(4)-Sextic

X(3)-Sextic

1

4

2 2 4 6

1, ,

2

4 2

2c

by a c s M

a

v y c y y y

c

Parameter free solutions

Z(4)-Sextic: P. Buganu, and R. Budaca, Phys. Rev. C 91 (2015) 014306.X(3)-Sextic: P. Buganu, and R. Budaca, J. Phys. G: Nucl. Part. Phys. 42 (2015) 105106.

Z(4)-SexticX(3)-Sextic

Experimental realisation of the predicted shape phase transitions

104 Ru

148 Nd196 Pt

120 Xe 126 Xe

130 Xe

196 Pt

Conclusions

Two new γ-rigid solutions have been proposed, called Z(4)-Sextic and X(3)-Sextic. For both of them, a sextic potential is used which leads to a quasi-exactly solvable equation.

Up to some scale parameters, the energies and the E2 transition probabilities depend on a single free parameter. For special cases when the term β2 or β4 cancels, parameter free solutions are obtained.

Varying the free parameter, shape phase transitions from an approximately spherical shape to a well deformed one are described. In the critical point the potential is flat leading to numerical results which are closed to those of X(3) and Z(4) for which an infinite square well was used.

In the critical point of X(3)-Sextic the states are approximately degenerate, indicating the presence of a symmetry which can offer answers for the unknown symmetry of X(5). The β bands of some X(5) candidate nuclei are well described in the present picture.

The plot of the free parameter as a function of the neutron number for isotopes of Xe, Pt, Sm and Nd reveales the presence of the proposed shape phase transitions in these chains.

Content

Introduction

Brief presentation of the new γ– rigid solutions

Numerical results

Conclusions

Introduction: Bohr Collective Model

The excitation spectra of the nuclei are interpreted as vibrations and rotations of the nuclear surface:

R0 – radius of spherical nucleus, αλμ – surface collective coordinates, Yλμ(θ,φ) – spherical harmonics.

Types of multipole deformations:

monopole dipole quadrupole octupole hexadecupole

00

, , 1 , ,R t R t Y

0 1 2

3 4

A. Bohr, Mat. Fyz. Medd. K. Dan. Vidensk. Selsk. 26 (1952) No. 14.A. Bohr, and B. R. Mottelson, Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 27 (1953) No. 16.

Quadrupole deformation: Wigner function

22 ' 2 1 2 3

'

: ; , ,RLab IntI I a D

0 0

5 2cos , 1,2,3.

4 3k kR R R R k k

0 spherical shape

0 deformed shape

20 2, 2 2, 2 2, 1 2, 1cos , sin , 0; 0 & 0,2 .2

a a a a a

Bohr-Mottelson transformation:

Euler angles

β=0.4 and γ=nπ/3 (n=0,1,2,3,4,5.): prolate(n=0,2,4), oblate (n=1,3,5)and triaxial in rest. L. Fortunato, Eur. Phys. J. A 26 (2005) 1-30.

The stretching of the nuclear axis. W. Greiner, J. A. Maruhn, Nuclear Models, Springer-Verlag Berlin Heidelberg (1996).

Page 15

Exactly separation of variables for γ=300

232 2 2 2 2

3 32 2 01

2

33 2 2 2

ˆ1 3 3 3ˆ ˆ ˆ ˆ, ( 1) ,24 4 4 4sin (30 )3

3( 1)1 2 24 , , = .

k

k

QQ Q Q Q L L R

k

L L Rd d B Bv v E

d d

Sextic oscillator with centrifugal barrier for the variable β

3

2

22

2 2

2 2 4 2 6

2

3( 1) 1

4

14 2 , M N

2

3 1 3( 1) 1 2 2

4 2 2

L L Rdv

d

v b a s M ab a

L L R s s

1

4, y= ab

a

Page 16

1.

2s M const c Condition to have a potential independent of state:

31

4 4 2

L LL R s M c

L – even

0

2

1

3

3, : ,0 , 1,4 . 2,8 ,...

2

, : , 2 , 1,6 , 2,10 ,... 2

7, : ,1 , 1,5 , 2,9 ,...

49

, : ,3 , 1,7 , 2,11 ,...4

K

K

K

K

M L K K K c K c

M L K K K c K c

M L K K K c K c

M L K K K c K c

L – odd

Final form of the potential

2 2 4 64 2 , m=0,1,2,3.K K Km m mv y c y y y u